I have this question (it's not a coursework question, I'm learning it by my own):(adsbygoogle = window.adsbygoogle || []).push({});

Use an appropiate power series expansion to find an asymptotic approximation as [tex]\epsilon[/tex] approaches 0+, correct to O(epsilon^2), for the two small roots of the equation: [tex]\epsilon x^3+x^2+2x-3=0[/tex] Then by using a suitable rescaling, find the first three terms of an asymptotic expansion as epsilon approaches 0+ of the singular root.

My way to go around this is to write down the next expansion series (because of the degree of the equation):

[tex]x(\epsilon)=\frac{1}{\epsilon^3}x_{-3}+ \frac{1}{\epsilon^2}x_{-2}+ \frac{1}{\epsilon^1}x_{-1}+x_0+x1\epsilon+...[/tex], the singular root is when epsilon equals null, i.e to solve x^2+2x-3=0=(x+3)(x-1) the roots are: x=-3 and x=1, I guess I need to choose the second root, now the recursive equation is:

[tex]x^2_{n+1}=-x_{n}/\epsilon-2/\epsilon+3/(x_{n}\epsilon)[/tex] Now, x0=1 and then use the iterative equation in order to find x_-3 and x_-2, am I correct or totally way off here?

Thanks in advance.

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# Asymptotic Analysis question.

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